We locate the centroid of a slice with <a href=fc.a68>this code</a>.
The argument to c is a vector v all of whose components must be positive.
If v is a vector then c(v) returns moments 0 and 1 of {x|x∙v&lt;1}∩[0, 1]<sup>n</sup>.
The centroid may be computed as (moment 1)/(moment 0).
This yields the coordinates of the centroid in the system of the preferred cube corner.
bt is the moments of the ‘big tent’.
<p>If σx is a permutation of the components of x and c(x) = (c, m) then c(σx) = (c, σm).
There is a floating point error in c proportional to 1/(product of components of argument).
c fails if any component of its argument is 0 but
<br>lim(x→0)c OF c((x, y, … w, x)) = c OF c((x, y, … w)) and
<br>lim(x→0)m OF c((x, y, … w, x)) = m OF c((x, y, … w)) with a c/2 appended to the vector.

<p>Of the arguments to c we say that x and y are complementary if
1/Σx<sub>i</sub> + 1/Σy<sub>i</sub> = 1.
The cube slices produced by complementary arguments form complements to the cube if you reflect one of them about the center of the cube.
If x is an argument to c then ip(x) is the complementary argument.
<p>From this equation:
<br> m'<sup>j</sup>
= <sub><font size=+3>∫</font></sub> (x<sup>j</sup> + d<sup>j</sup>)<i>dm</i>
= m<sup>j</sup> + md<sup>j</sup>
<br>from <a href=../Tumble/tensor.html>here</a> we conclude that the vector sum of the moment 1’s should be (½, ½, …).
Consider the test ts((3, 0.001)) and corresponding test.
That divides the square very nearly into <img src=img.svg />.
<pre>centroid 1    m1/m0
centroid 2    (1, 1, …) - m1'/m0'
Weighted mean at (½, ½, …)

(m1/m0)m0 + ((1, 1, …) - m1'/m0')m0' = (½, ½, …)
m1 + (m0', m0', …) - m1'</pre>
<tt>mq</tt> is the location of the centroid of combined slices a and b when b is reflected thru the cube center.
I take it as good evidence that the routine c is correct that “Foul cent” is not printed.
<p><tt>sl</tt> is the fraction of the vertical body diagonal above the water.
<p>Code constructs beginning “(0=1 |” are code fragments that are turned off.
Changing them to begin “(0=0 |” turns them on.
Each is a test of sorts.
<p>The three columns are:<ul>
<li>Cube density
<li>Height above water of highest point of body.
<li>Height of centroid of that part of body above water.</ul>
Other places we consider the moment above the water which is ½ the height of centroid when density = ½.
This confusion caused a day of debugging.